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Article A boundary value problem for second-order nonlinear difference equations on the integers(Cambridge Univ Press, 2005) Dal, F; Guseinov, GSIn this study, we are concerned with a boundary value problem (BVP) for nonlinear difference equations on the set of all integers Z, under the assumption that the left-hand side is a second-order linear difference expression which belongs to the so-called Weyl-Hamburger limit-circle case. The BVP is considered in the Hilbert space l(2) and includes boundary conditions at infinity. Existence and uniqueness results for solution of the considered BVP are established.Article Citation - WoS: 44Citation - Scopus: 46Integrable Equations on Time Scales -: Art. No. 113510(Amer inst Physics, 2005) Gürses, M; Guseinov, GS; Silindir, BIntegrable systems are usually given in terms of functions of continuous variables (on R), in terms of functions of discrete variables (on Z), and recently in terms of functions of q-variables (on K-q). We formulate the Gel'fand-Dikii (GD) formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations. In particular they yield integrable equations over integers (difference equations) and over q-numbers (q-difference equations). We formulate the GD formalism also in terms of shift operators for all regular-discrete time scales. We give a method allowing to construct the recursion operators for integrable systems on time scales. Finally, we give a trace formula on time scales and then construct infinitely many conserved quantities (Casimirs) of the integrable systems on time scales. (c) 2005 American Institute of Physics.
